Textbook Question
55–58. Marginal cost Consider the following marginal cost functions.
a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.
C′(x) = 300+10x−0.01x²
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Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.1.47a
Depletion of natural resources Suppose r(t) = r0e^−kt, with k>0, is the rate at which a nation extracts oil, where r0=10⁷ barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2×10⁹ barrels.
a. Find Q(t), the total amount of oil extracted by the nation after t years.
Verified step by step guidance1
Identify the given rate of extraction function: \(r(t) = r_0 e^{-kt}\), where \(r_0 = 10^7\) barrels/year and \(k > 0\).
Recall that \(Q(t)\), the total amount of oil extracted after \(t\) years, is the integral of the rate function \(r(t)\) from 0 to \(t\):
\(Q(t) = \int_0^t r(\tau) \, d\tau\).
Substitute the given rate function into the integral:
\(Q(t) = \int_0^t r_0 e^{-k\tau} \, d\tau\).
Integrate the exponential function with respect to \(\tau\):
\(\int e^{-k\tau} \, d\tau = -\frac{1}{k} e^{-k\tau} + C\).
Apply the definite integral limits from 0 to \(t\) and multiply by \(r_0\):
\(Q(t) = r_0 \left[-\frac{1}{k} e^{-k\tau} \right]_0^t = r_0 \left(-\frac{1}{k} e^{-kt} + \frac{1}{k} e^{0} \right)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Decay Function
An exponential decay function models quantities that decrease at a rate proportional to their current value. In this problem, the extraction rate r(t) = r0e^(-kt) shows how the oil extraction rate decreases over time, with k > 0 controlling the speed of decay.
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Exponential Growth & Decay
Definite Integral for Accumulated Quantity
To find the total amount extracted over time, we integrate the rate function r(t) from 0 to t. The definite integral sums the instantaneous extraction rates, giving the cumulative quantity Q(t) extracted after t years.
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Initial Conditions and Parameters
Understanding the given constants, such as the initial extraction rate r0 and the total reserve, is crucial. These parameters set the scale and constraints for the problem, ensuring the solution is physically meaningful and consistent with the resource limits.
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Related Practice
Textbook Question
6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.
Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.
a. What is the radius of a cylindrical shell at a point x in [0, 4]?
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Textbook Question
{Use of Tech} Oscillating motion A mass hanging from a spring is set in motion, and its ensuing velocity is given by v(t) = 2π cos πt, for t≥0. Assume the positive direction is upward and s(0)=0.
a. Determine the position function, for t≥0.
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Textbook Question
Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).
a. How much work is required to pump the water out of the trough (to the level of the top of the trough) when it is full?
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Textbook Question
A vertical spring A 10-kg mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx.
a. How much work is required to compress the spring and lift the mass 0.5 m?
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Textbook Question
17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.
a. Determine the position function, for t≥0, using the antiderivative method
v(t) = 6−2t on [0, 5]; s(0)=0
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